We perform a systematic study of the lattice dynamics and the lattice thermal conductivity, kappa, of monolayer group 13 monochalcogenides MX (M = Ga, In; X = S, Se, Te) by combining an iterative solution for linearized phonon Boltzmann transport equation and density functional theory. Among the competing factors influencing kappa, harmonic parameters along with the atomic masses dominate over anharmonicity. An increase in atomic mass leads to a decrease in phonon frequencies and phonon group velocities and consequently in kappa. At T = 300 K, the calculated kappa values are 54.9, 48.1, 44.3, 25.0, 22.3, and 17.3 W m(-1) K-1 for GaS, InS, GaSe, InSe, GaTe, and InTe monolayers, respectively. Further analysis of anharmonic scattering rates and average scattering matrix elements evidences that the anharmonicity characterized by the third-order IFCs in GaS and InS are the largest among all monolayer group 13 monochalcogenides despite the largest kappa values. This is attributed to a strong interaction between nonbonding lone-pair s electrons around the S atom and adjacent bonding electrons. In addition, the kappa of these monolayers further reduces to 50% for sample sizes 300-400 nm. Our findings provide fundamental insights into thermal transport in monolayer group 13 monochalcogenides and should stimulate further experimental exploration of thermal transport in these materials for possible theromoelectric and thermal management applications.